The

Archimedean property of the

real numbers can be stated as follows: for any real number

**x**, there exists an integer greater than

**x**. (In other words, the set of integers is not

bounded above.) A

corollary of this fact, which is itself sometimes called the "Archimedean property", is that for any positive

reals

**x** and

**y**, there exists an

integer **n** such that

**n x** is greater than

**y**.

This property of the reals can be used to prove that there is a rational number between any two reals.

Archimedes' property is a consequence of the completeness axiom (or least upper bound property) for the real numbers, which states that any non-empty subset of the reals which is bounded above has a supremum. A proof of the property runs as follows: assume that there is some real number **x** such that no integer is greater than **x**. Then **x** is an upper bound for the integers, so by the completeness axiom the set of integers has a supremum--call it **s**. Because **s** is a least upper bound, **s-1** cannot be an upper bound for the integers; hence there is an integer **n** which is greater than **s-1**. But then **n+1** is an integer greater than **s**, a contradiction.